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E-grāmata: Fractional Order Analysis: Theory, Methods and Applications

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  • Formāts: PDF+DRM
  • Izdošanas datums: 17-Aug-2020
  • Izdevniecība: John Wiley & Sons Inc
  • Valoda: eng
  • ISBN-13: 9781119654209
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Fractional Order Analysis: Theory, Methods and ApplicationsPalielināt
  • Formāts: PDF+DRM
  • Izdošanas datums: 17-Aug-2020
  • Izdevniecība: John Wiley & Sons Inc
  • Valoda: eng
  • ISBN-13: 9781119654209
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"The book contains new research findings in fractional order analysis. Different tools and techniques from the field of fractional order analysis are found to be extremely helpful in practical applications because such analyses can help in developing mathematical methods or models more accurately. The book covers topics such as: conformable fractional operators, local fractional proportional derivatives, fractional derivative and integral operators, different inequalities via several fractional environments, different type of fractional integral operators, fractional-order chaotic model, flow models in fractional environment, multi-dimensional fractional order models, fractional modelling of the transmission dynamics of Influenza, and Zika virus model with various fractional derivatives. Using this book, readers can find several useful, relevant and connected topics in one place, which is necessary for crucial understanding of research problems of an applied nature"--

A guide to the new research in the field of fractional order analysis 

Fractional Order Analysis contains the most recent research findings in fractional order analysis and its applications. The authors—noted experts on the topic—offer an examination of the theory, methods, applications, and the modern tools and techniques in the field of fractional order analysis. The information, tools, and applications presented can help develop mathematical methods and models with better accuracy. 

Comprehensive in scope, the book covers a range of topics including: new fractional operators, fractional derivatives, fractional differential equations, inequalities for different fractional derivatives and fractional integrals, fractional modeling related to transmission of Malaria, and dynamics of Zika virus with various fractional derivatives, and more. Designed to be an accessible text, several useful, relevant and connected topics can be found in one place, which is crucial for an understanding of the research problems of an applied nature. This book:  

  • Contains recent development in fractional calculus 
  • Offers a balance of theory, methods, and applications 
  • Puts the focus on fractional analysis and its interdisciplinary applications, such as fractional models for biological models 
  • Helps make research more relevant to real-life applications 

Written for researchers, professionals and practitioners, Fractional Order Analysis offers a comprehensive resource to fractional analysis and its many applications as well as information on the newest research.  

Preface xi
List of Contributors
xv
About the Editors xix
1 On the Fractional Derivative and Integral Operators
1(42)
Mustafa A. Dokuyucu
1.1 Introduction
1(1)
1.2 Fractional Derivative and Integral Operators
2(15)
1.2.1 Properties of the Griinwald-Letnikov Fractional Derivative and Integral
2(4)
1.2.1.1 Integral of Arbitrary Order
6(1)
1.2.1.2 Derivatives of Arbitrary Order
7(2)
1.2.2 Properties of Riemann-Liouville Fractional Derivative and Integral
9(1)
1.2.2.1 Unification of Integer-Order Derivatives and Integrals
10(2)
1.2.2.2 Integrals of Arbitrary Order
12(2)
1.2.2.3 Derivatives of Arbitrary Order
14(3)
1.3 Properties of Caputo Fractional Derivative and Integral
17(3)
1.4 Properties of the Caputo-Fabrizio Fractional Derivative and Integral
20(4)
1.5 Properties of the Atangana-Baleanu Fractional Derivative and Integral
24(4)
1.6 Applications
28(15)
1.6.1 Keller-Segel Model with Caputo Derivative
28(1)
1.6.1.1 Existence and Uniqueness Solutions
28(3)
1.6.1.2 Uniqueness of Solution
31(1)
1.6.1.3 Keller-Segel Model with Atangana-Baleanu Derivative in Caputo Sense
32(1)
1.6.1.4 Uniqueness of Solution
33(1)
1.6.2 Cancer Treatment Model with Caputo-Fabrizio Fractional Derivative
34(1)
1.6.2.1 Existence Solutions
35(3)
1.6.2.2 Uniqueness Solutions
38(1)
1.6.2.3 Conclusion
39(1)
Bibliography
40(3)
2 Generalized Conformable Fractional Operators and Their Applications
43(48)
Muhammad Adil Khan
Tahir Ullah Khan
2.1 Introduction and Preliminaries
43(3)
2.2 Generalized Conformable Fractional Integral Operators
46(6)
2.2.1 Construction of New Integral Operators
47(5)
2.3 Generalized Conformable Fractional Derivative
52(8)
2.4 Applications to Integral Equations and Fractional Differential Equations
60(3)
2.4.1 Equivalence Between the Generalized Nonlinear Problem and the Volterra Integral Equation
61(1)
2.4.2 Existence and Uniqueness of Solution for the Nonlinear Problem
61(2)
2.5 Applications to the Field of Inequalities
63(28)
2.5.1 Inequalities Related to the Left Side of Hermite-Hadamard Inequality
65(9)
2.5.1.1 Applications to Special Means of Real Numbers
74(1)
2.5.1.2 Applications to the Midpoint Formula
75(1)
2.5.2 Inequalities Related to the Right Side of Hermite-Hadamard Inequality
76(8)
2.5.2.1 Applications to Special Means of Real Numbers
84(1)
2.5.2.2 Applications to the Trapezoidal Formula
84(2)
Bibliography
86(5)
3 Analysis of New Trends of Fractional Differential Equations
91(22)
Abdon Atangana
Ali Akgul
3.1 Introduction
91(1)
3.2 Theory
92(9)
3.3 Discretization
101(2)
3.4 Experiments
103(1)
3.5 Stability Analysis
104(6)
3.6 Conclusion
110(3)
Bibliography
111(2)
4 New Estimations for Exponentially Convexity via Conformable Fractional Operators
113(20)
Alper Ekinci
Sever S. Dragomir
4.1 Introduction
113(4)
4.2 Main Results
117(16)
Bibliography
130(3)
5 Lyapunov-type Inequalities for Local Fractional Proportional Derivatives
133(18)
Thabet Abdeljawad
5.1 Introduction
133(2)
5.2 The Local Fractional Proportional Derivatives and Their Generated Nonlocal Fractional Proportional Integrals and Derivatives
135(2)
5.3 Lyapunov-Type Inequalities for Some Nonlocal and Local Fractional Operators
137(4)
5.4 The Lyapunov Inequality for the Sequential Local Fractional Proportional Boundary Value Problem
141(3)
5.5 A Higher-Order Extension of the Local Fractional Proportional Operators and an Associate Lyapunov Open Problem
144(2)
5.6 Conclusion
146(5)
Acknowledgement
146(1)
Bibliography
147(4)
6 Minkowski-Type Inequalities for Mixed Conformable Fractional Integrals
151(18)
Erhan Set
Muhamet E. Ozdemir
6.1 Introduction and Preliminaries
151(7)
6.2 Reverse Minkowski Inequality Involving Mixed Conformable Fractional Integrals
158(2)
6.3 Related Inequalities
160(9)
Bibliography
167(2)
7 New Estimations for Different Kinds of Convex Functions via Conformable Integrals and Riemann-Liouville Fractional Integral Operators
169(26)
Ahmet Ocak Akdemir
Hemen Dutta
7.1 Introduction
169(3)
7.2 Some Generalizations for Geometrically Convex Functions
172(7)
7.3 New Inequalities for Co-ordinated Convex Functions
179(16)
Bibliography
191(4)
8 Legendre-Spectral Algorithms for Solving Some Fractional Differential Equations
195(30)
Youssri H. Youssri
Waleed M. Abd-Elhameed
8.1 Introduction
195(2)
8.2 Some Properties and Relations Concerned with Shifted Legendre Polynomials
197(3)
8.3 Galerkin Approach for Treating Fractional Telegraph Type Equation
200(4)
8.4 Discussion of the Convergence and Error Analysis of the Suggested Double Expansion
204(3)
8.5 Some Test Problems for Fractional Telegraph Equation
207(2)
8.6 Spectral Algorithms for Treating the Space Fractional Diffusion Problem
209(5)
8.6.1 Transformation of the Problem
210(1)
8.6.2 Basis Functions Selection
211(2)
8.6.3 A Collocation Scheme for Solving Eq. 8.44
213(1)
8.6.4 An Alternative Spectral Petrov-Galerkin Scheme for Solving Eq. (8.44)
214(1)
8.7 Investigation of Convergence and Error Analysis
214(2)
8.8 Numerical Results and Comparisons
216(4)
8.9 Conclusion
220(5)
Bibliography
220(5)
9 Mathematical Modeling of an Autonomous Nonlinear Dynamical System for Malaria Transmission Using Caputo Derivative
225(28)
Abdon Atangana
Sania Qureshi
9.1 Introduction
225(2)
9.2 Mathematical Preliminaries
227(1)
9.3 Model Formulation
228(2)
9.4 Basic Properties of the Fractional Model
230(3)
9.4.1 Reproductive Number
230(1)
9.4.2 Existence and Stability of Disease-free Equilibrium Points
231(1)
9.4.3 Existence and Stability of Endemic Equilibrium Point
232(1)
9.5 Existence and Uniqueness of the Solutions
233(4)
9.5.1 Positivity of the Solutions
236(1)
9.6 Numerical Simulations
237(10)
9.7 Conclusion
247(6)
Bibliography
250(3)
10 MHD-free Convection Flow Over a Vertical Plate with Ramped Wall Temperature and Chemical Reaction in View of Nonsingular Kernel
253(30)
Muhammad B. Riaz
Abdon Atangana
Syed T. Saeed
10.1 Introduction
253(1)
10.2 Mathematical Model
254(2)
10.2.1 Preliminaries
256(1)
10.3 Solution
256(7)
10.3.1 Concentration Fields
257(1)
10.3.1.1 Concentration Field with Caputo Time-Fractional Derivative
257(1)
10.3.1.2 Concentration Field with Caputo-Fabrizio Time-Fractional Derivative
257(1)
10.3.1.3 Concentration Field with Atangana-Baleanu Time-Fractional Derivative
257(1)
10.3.2 Temperature Fields
258(1)
10.3.2.1 Temperature Field with Caputo Time-Fractional Derivative
258(1)
10.3.2.2 Temperature Field with Caputo-Fabrizio Time-Fractional Derivative
258(1)
10.3.2.3 Temperature Field with Atangana-Baleanu Time-Fractional Derivative
258(1)
10.3.3 Velocity Fields
259(1)
10.3.3.1 Velocity Field with Caputo Time-Fractional Derivative
259(1)
10.3.3.2 Velocity Field with Caputo-Fabrizio Time-Fractional Derivative
259(3)
10.3.3.3 Velocity Field with Atangana-Baleanu Time-Fractional Derivative
262(1)
10.4 Results and Discussion
263(1)
10.5 Conclusion
263(20)
Bibliography
279(4)
11 Comparison of the Different Fractional Derivatives for the Dynamics of Zika Virus
283(24)
Muhammad Altaf Khan
11.1 Introduction
283(1)
11.2 Background of Fractional Operators
284(2)
11.3 Model Framework
286(1)
11.4 A Fractional Zika Model with Different Fractional Derivatives
287(1)
11.5 Numerical Scheme for Caputo-Fabrizio Model
288(5)
11.5.1 Solutions Existence for the Atangana-Baleanu Model
289(2)
11.5.2 Numerical Scheme for Atangana-Baleanu Model
291(2)
11.6 Numerical Results
293(10)
11.7 Conclusion
303(4)
Bibliography
303(4)
Index 307
HEMEN DUTTA, PHD, is Faculty Member in the Department of Mathematics at Gauhati University, Guwahati, India.

AHMET OCAK AKDEMIR, PHD, is Associate Professor, Ar brahim Ēeēen University, Faculty of Science and Letters, Department of Mathematics, Ar, Turkey.

ABDON ATANGANA, PHD, is Professor, Institute for Groundwater Studies, University of the Free State, Bloemfontein, South Africa.
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